which implies that each \(X_{m,j}^{(b)}\)  is a separable sub-Gaussian process under the semi-metric \(d(\theta _1,\theta _2) = [\frac{1}{m} \sum _{i=1}^m F_q(z_i)^2]^{1/2}\Vert \theta _1-\theta _2\Vert \) . Using Corollary 2.2.8 in [1]}, we have for some universal constant \(K\) :
\( \mathbb {E}^\star \left( \sup _{\theta \in \Theta } |X_{m,j}^{(b)}(\theta )| \right) \le \mathbb {E}^\star \left( |X_{m,j}^{(b)}(\theta ^\dagger )| \right) + K \int _0^\infty \sqrt{\log D(\varepsilon ,d)}d\varepsilon , \) 
